When Neil J. McNeill Jr., principal of the Middle School for Art and Philosophy in Brooklyn, learned that fewer than 4 percent of his students had passed state exams in math last year, he was frustrated.

It so happened that he shared a building with one of the top-performing schools in the Brownsville neighborhood, Kings Collegiate Charter School, where 37 percent of the students had passed, well above the New York City middle-school average of 27 percent.

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“We are kind of two ships in the night,” Mr. McNeill, 39, said recently.

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But two decades since the schools began to appear, educators from both systems concede that very little of what has worked for charter schools has found its way into regular classrooms. Testy political battles over space and money, including one that became glaringly public in New York State this spring, have inhibited attempts at collaboration. The sharing of school buildings, which in theory should foster communication, has more frequently led to conflict.

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In recent years, educational leaders, concerned about hostilities between the two types of schools, have worked to encourage warmer relations. In Tulsa, Okla., charter schools and district schools are working together to improve teaching quality. And in Spring Branch, Tex., charter school leaders are helping train district teachers and principals.

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“It’s like putting a Burger King kitty-corner to a McDonald’s and expecting — in the same location and competing for the same families — warm and fuzzy cooperation,” said Bruce Fuller, a professor of education and public policy at the University of California, Berkeley.

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The Bronx Charter School for Excellence and the nearby Public School 85 are exceptions. Since winning a state grant in 2013, the schools have worked together to improve the quality of their kindergarten programs. They have held teacher exchanges, and every week they come together to discuss topics like student participation and reading strategies.

At first, staff members on both sides seemed dubious, and sat on different sides of the room.

“There was this idea that we were trying to turn this school into the charter, this grand conspiracy of, ‘Oh my God, are we going to be taken over, are we going to be closed?’ ” said Charlene Reid, head of the Bronx Charter School for Excellence.

But now, staff members said they shared observations and ideas by email several times per day. As a result, P.S. 85 began assessing student reading levels earlier in the year and increased the rigor of reading assignments.

In the four states that released results from their annual statewide assessments–Missouri, West Virginia, Oregon, and Washington–students exceeded expectations on the Smarter Balanced Assessment (SBA),…